. Differential and integral calculus, an introductory course for colleges and engineering schools. chorder the three successive integrations are performed, care beingtaken in each case to employ proper limits. Example. Let us find the value of the triple integral J J j xdzdy dx, v where V is the space included in an octant of the sphere x2 + y1 + z1 = Solution. We cut V into slices by planes parallel to the yz-p\ the triple integral I, we have x=r y= v r2—x2 z= V?i—xi—y2 I = j J j x dz dy dx, x=0 y=0 z=0 the limits of integration being determined as follows:In the first int
. Differential and integral calculus, an introductory course for colleges and engineering schools. chorder the three successive integrations are performed, care beingtaken in each case to employ proper limits. Example. Let us find the value of the triple integral J J j xdzdy dx, v where V is the space included in an octant of the sphere x2 + y1 + z1 = Solution. We cut V into slices by planes parallel to the yz-p\ the triple integral I, we have x=r y= v r2—x2 z= V?i—xi—y2 I = j J j x dz dy dx, x=0 y=0 z=0 the limits of integration being determined as follows:In the first integration j x dz, we hold x and y constant and extend the §219 MULTIPLE INTEGRALS 335 summation along the prism AB fromz = 0 at A to z = Vr2-x2-y2 at the second integration we hold xconstant (z does not occur in the in-tegrand) and thus extend the sum-mation over all the prisms in theslice MPBN from y = 0 at M to y-a yV2 — x2 at JV. The final integra-tion extends the summation over allthe slices of the octant from x = 0 N at 0 to x = r at C The numerical calculation is as follows;. v^~. J xdz — x V? 3 = 0 Then by the table of integrals 2/= Vr2_a;2 Vr2 — X2 1/2^ ?/ W—x2—z/2-f- (r2 — a;2) sin-1 Z/=0 Finally, Vi Jy=o (r2 — z2):c. 7 . | J> _ ^ & _ _ x |> _ ^J. g. A ns. Second Solution. Cut the solid into slices by planes parallel to the:n/-plane. Let the student draw the figure and complete the solution. 219. The Triple Integral in Polar Coordinates. The posi-tion of a point P may be givenby three coordinates, p, 6, 4>, asshown in the figure. We mayregard p as the radius of thesphere on which P lies and whosecenter is at 0. 6 is the longitudeof P, and 4> is its spherical coordinates, asthey are called, are connectedwith the Cartesian coordinates
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